Deep learning has emerged as a dominant approach in machine learning and has achieved remarkable success in various domains such as computer vision and natural language processing. Its influence has progressively extended to numerous research areas within the fields of science and engineering. In this presentation, I will outline our work on the design and training of a foundation model, named PDEformer, which aims to serve as a flexible and efficient solver across a spectrum of parametric PDEs. PDEformer is specifically engineered to facilitate a range of downstream tasks, including but not limited to parameter estimation and system identification. Its design is tailored to accommodate applications necessitating repetitive solving of PDEs, where a balance between efficiency and accuracy is sought. This is a joint workshop with the Serabol program.

This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.

"Data driven governing equations approximation using deep neural networks", Journal of Computational Physics (2019) will be discussed in this Journal Club. We present a numerical framework for approximating unknown governing equations using observation data and deep neural networks (DNN). In particular, we propose to use residual network (ResNet) as the basic building block for equation approximation. We demonstrate that the ResNet block can be considered as a one-step method that is exact in temporal integration. We then present two multi-step methods, recurrent ResNet (RT-ResNet) method and recursive ReNet (RS-ResNet) method. The RT-ResNet is a multi-step method on uniform time steps, whereas the RS-ResNet is an adaptive multi-step method using variable time steps. All three methods presented here are based on integral form of the underlying dynamical system. As a result, they do not require time derivative data for equation recovery and can cope with relatively coarsely distributed trajectory data. Several numerical examples are presented to demonstrate the performance of the methods. If you want to participate in the seminar, you need to enter IBS builiding (https://www.ibs.re.kr/bimag/visiting/). Please contact if you first come IBS to get permission to enter IBS building.

We discuss how optimal transport, which is a theory for matching different distributions in a cost effective way, is applied to the supercooled Stefan problem, a free boundary problem that describes the interface dynamics of supercooled water freezing into ice. This problem exhibits a highly unstable behaviour and its mathematical study has been limited mostly to one space dimension, and widely open for multi-dimensional cases. We consider a version of optimal transport problem that considers stopping of the Brownian motion, whose solution is then translated into a solution to the supercooled Stefan problem in general dimensions.

In this talk, we consider the Boltzmann equation in general 3D toroidal domains with a specular reflection boundary condition. So far, it is a well-known open problem to obtain the low-regularity solution for the Boltzmann equation in general non-convex domains because there are grazing cases, such as inflection grazing. Thus, it is important to analyze trajectories which cause grazing. We will provide new analysis to handle these trajectories in general 3D toroidal domains.

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심사위원장: 김재경, 심사위원:김용정, 정연승, 김진수(포항공과대학교), 이승규(고려대학교)

(John Hubbard's Mini-course: Lecture 3)

Post-critically finite (PCF) rational maps are a fascinating class of dynamical systems with rich mathematical structures. In this minicourse, we explore the interplay between topology, geometry, and dynamics in the study of PCF rational maps. [Lecture 3: Geometry of PCF rational maps] Geometry of PCF rational maps The topological models for PCF rational maps we discuss define canonical quasi-symmetric classes of metrics on their Julia sets. We investigate the conformal dimensions of Julia sets, which measure their geometric complexity and provide insights into the underlying dynamics. Through this exploration, we uncover the intricate relationship between the topology, geometry, and dynamics of PCF rational maps.

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심사위원장: 황강욱, 심사위원: 강완모, 김용정, 전현호, 문일철(산업및시스템공학과)

Weak discuss existence of singular solutions for Stokes and the Navier-Stokes equations in the half-space. We construct their solutions whose normal derivatives are unbounded for the Stokes and Navier-Stokes equations near boundary away from support of singular data. 

The finite quotient groups of étale fundamental groups of algebraic curves in positive characteristic are precisely determined, but without explicit construction of quotient maps, by well-known results of Raynaud, Harbater and Pop, previously known as Abhyankar's conjecture. Katz, Rojas León and Tiep have been studying the constructive side of this problem using certain "easy to remember" local systems. In this talk, I will discuss the main results and methods of this project in the case of a specific type of local systems called hypergeometric sheaves.

Two-way online correlated selection (two-way OCS) is an online algorithm that, at each timestep, takes a pair of elements from the ground set and irrevocably chooses one of the two elements, while ensuring negative correlation in the algorithm's choices. OCS was initially invented by Fahrbach, Huang, Tao, and Zadimoghaddam (FOCS 2020, JACM 2022) to break a natural long-standing barrier in edge-weighted online bipartite matching. They posed two open questions, one of which was the following: Can we obtain n-way OCS for $n >2$, in which the algorithm can be given $n >2$ elements to choose from at each timestep? In this talk, we affirmatively answer this open question by presenting a three-way OCS which is simple to describe: it internally runs two instances of two-way OCS, one of which is fed with the output of the other. Contrast to its simple construction, we face a new challenge in analysis that the final output probability distribution of our three-way OCS is highly elusive since it requires the actual output distribution of two-way OCS. We show how we tackle this challenge by approximating the output distribution of two-way OCS by a flatter distribution serving as a safe surrogate. This is joint work with Hyung-Chan An.

(John Hubbard's Mini-course: Lecture 2)

Post-critically finite (PCF) rational maps are a fascinating class of dynamical systems with rich mathematical structures. In this minicourse, we explore the interplay between topology, geometry, and dynamics in the study of PCF rational maps. [Lecture 2: Topology of PCF rational maps] W.Thurston's and D.Thurston's characterizations provide powerful frameworks for understanding the topological dynamics of rational maps. We delve into these characterizations, exploring their implications for the dynamics of PCF rational maps. Additionally, we discuss finite subdivision rules and topological surgeries, such as matings, tunings, and decompositions, as tools for constructing and analyzing PCF rational maps in topological ways.

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심사위원장: 강완모, 심사위원: 김동환, 황강욱, 윤세영(김재철AI대학원), 민승기(산업및시스템공학과)

Quantum embedding is a fundamental prerequisite for applying quantum machine learning techniques to classical data, and has substantial impacts on performance outcomes. In this study, we present Neural Quantum Embedding (NQE), a method that efficiently optimizes quantum embedding beyond the limitations of positive and trace-preserving maps by leveraging classical deep learning techniques. NQE enhances the lower bound of the empirical risk, leading to substantial improvements in classification performance. Moreover, NQE improves robustness against noise. To validate the effectiveness of NQE, we conduct experiments on IBM quantum devices for image data classification, resulting in a remarkable accuracy enhancement. In addition, numerical analyses highlight that NQE simultaneously improves the trainability and generalization performance of quantum neural networks, as well as of the quantum kernel method.

We provide general upper and lower bounds for the Gromov–Hausdorff distance d_GH(S^m, S^n) between spheres S^m and S^n (endowed with the round metric) for 0 <= m < n <= 1. Some of these lower bounds are based on certain topological ideas related to the Borsuk–Ulam theorem. Via explicit constructions of (optimal) correspondences, we prove that our lower bounds are tight in the cases of d_GH(S^0, S^n), d_GH(S^m, S^\infty), d_GH(S^1, S^2), d_GH(S^1, S^3), and d_GH(S^2, S^3). We also formulate a number of open questions.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.

"PenDA, a rank-based method for personalized differential analysis: Application to lung cancer", Plos Comp. Biol. (2020) will be discussed in this Journal Club. The hopes of precision medicine rely on our capacity to measure various high-throughput genomic information of a patient and to integrate them for personalized diagnosis and adapted treatment. Reaching these ambitious objectives will require the development of efficient tools for the detection of molecular defects at the individual level. Here, we propose a novel method, PenDA, to perform Personalized Differential Analysis at the scale of a single sample. PenDA is based on the local ordering of gene expressions within individual cases and infers the deregulation status of genes in a sample of interest compared to a reference dataset. Based on realistic simulations of RNA-seq data of tumors, we showed that PenDA outcompetes existing approaches with very high specificity and sensitivity and is robust to normalization effects. Applying the method to lung cancer cohorts, we observed that deregulated genes in tumors exhibit a cancer-type-specific commitment towards up- or down-regulation. Based on the individual information of deregulation given by PenDA, we were able to define two new molecular histologies for lung adenocarcinoma cancers strongly correlated to survival. In particular, we identified 37 biomarkers whose up-regulation lead to bad prognosis and that we validated on two independent cohorts. PenDA provides a robust, generic tool to extract personalized deregulation patterns that can then be used for the discovery of therapeutic targets and for personalized diagnosis. An open-access, user-friendly R package is available at https://github.com/bcm-uga/penda. If you want to participate in the seminar, you need to enter IBS builiding (https://www.ibs.re.kr/bimag/visiting/). Please contact if you first come IBS to get permission to enter IBS building.

(Dynamics Seminar)

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심사위원장: 권순식, 심사위원: 강문진, 변재형, 배명진, 조용근(전북대학교)

Reinforcement learning (RL) has become one of the most central problems in machine learning, showcasing remarkable success in recommendation systems, robotics and super-human level game plays. Yet, existing literature predominantly focuses on (almost) fully observable environments, overlooking the complexities of real-world scenarios where crucial information remains hidden. In this talk, we consider reinforcement learning in partially observable systems through the proposed framework of the Latent Markov Decision Process (LMDP). In LMDPs, an MDP is randomly drawn from a set of possible MDPs at the beginning of the interaction, but the context -- the latent factors identifying the chosen MDP -- is not revealed to the agent. This opacity poses new challenges for decision-making, particularly in scenarios like recommendation systems without sensitive user data, or medical treatments for undiagnosed illnesses. Despite the significant relevance of LMDPs to real-world problems, existing theories rely on restrictive separation assumptions -- an unrealistic constraint in practical applications. We present a series of new results addressing this gap: from leveraging higher-order information to develop sample-efficient RL algorithms, to establishing lower bounds and improved results under more realistic assumptions within Latent MDPs.

(John Hubbard's Mini-course: Lecture 1)

Post-critically finite (PCF) rational maps are a fascinating class of dynamical systems with rich mathematical structures. In this minicourse, we explore the interplay between topology, geometry, and dynamics in the study of PCF rational maps. [Lecture 1: What are PCF rational maps?] We begin by introducing PCF rational maps, highlighting their significance in complex dynamics.

This presentation focuses on unbiased simulation methods for quantities associated with sample paths from stochastic differential equations. Unbiased simulation methods can be found by changing probability measures with appropriately selecting the Radon-Nikodym derivative processes. I propose an unbiased Monte-Carlo simulation method that can be used even when the Girsanov kernel for the change of probability measures is not bounded. Then, I illustrate its practical application through an example involving unbiased Monte Carlo simulation for pricing the continuously averaging arithmetic Asian options under the Black-Scholes model.

Very little is known about critical properties of graphs in the hierarchy of monotone classes, i.e. classes closed under taking (not necessarily induced) subgraphs. We distinguish four important levels in this hierarchy and discuss possible new levels by focusing on the Hamiltonian cycle problem. In particular, we obtain a number of results for this problem on monotone classes.

This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.

"Optimal-Transport Analysis of Single-Cell Gene Expression Identifies Developmental Trajectories in Reprogramming", Cell (2019) will be discussed in this Journal Club. Understanding the molecular programs that guide differentiation during development is a major challenge. Here, we introduce Waddington-OT, an approach for studying developmental time courses to infer ancestor-descendant fates and model the regulatory programs that underlie them. We apply the method to reconstruct the landscape of reprogramming from 315,000 single-cell RNA sequencing (scRNA-seq) profiles, collected at half-day intervals across 18 days. The results reveal a wider range of developmental programs than previously characterized. Cells gradually adopt either a terminal stromal state or a mesenchymal-to-epithelial transition state. The latter gives rise to populations related to pluripotent, extra-embryonic, and neural cells, with each harboring multiple finer subpopulations. The analysis predicts transcription factors and paracrine signals that affect fates and experiments validate that the TF Obox6 and the cytokine GDF9 enhance reprogramming efficiency. Our approach sheds light on the process and outcome of reprogramming and provides a framework applicable to diverse temporal processes in biology. If you want to participate in the seminar, you need to enter IBS builiding (https://www.ibs.re.kr/bimag/visiting/). Please contact if you first come IBS to get permission to enter IBS building.

Given a smooth manifold or orbifold M and a Lie group G acting transitively on a space X, we consider the space of all (G, X)-structures on M up to an appropriate equivalence relation. This space, known as the deformation space of (G, X)-structures on M, encodes information about how one can "deform" the (G, X)-manifold M. In this talk, I will provide a general definition of deformation spaces and character varieties, which capture the local structure of the deformation space. Additionally, I will introduce a class of orbifolds called the Coxeter orbifolds, for which deformation spaces can be computed using an approach due to the foundational work of E. Vinberg.

1. 데이터 분석 업무의 이해(김준범)- 데이터 분석가의 역할 소개 2. 초거대 언어 모델 동향(김정섭)-GPT-3 부터 Llama-3까지 이미 우리 삶 속에 깊숙이 자리잡은 초거대 언어 모델의 동향 3. 데이터 분석가에서 공직으로 오게된 과정과 앞으로의 계획(심규석)- 삼성화재에서의 데이터 분석 및 AI 모델링 업무, 행정안전부에서의 데이터 분석과제 기획·관리 및 공무원의 데이터 분석 역량지원 업무 전반에 관한 설명과 함께 각 기관을 지원하게 된 동기, 지원방법, 준비사항 등

A cross-cap drawing of a graph G is a drawing on the sphere with g distinct points, called cross-caps, such that the drawing is an embedding except at the cross-caps, where edges cross properly. A cross-cap drawing of a graph G with g cross-caps can be used to represent an embedding of G on a non-orientable surface of genus g. Mohar conjectured that any triangulation of a non-orientable surface of genus g admits a cross-cap drawing with g cross-caps in which each edge of the triangulation enters each cross-cap at most once. Motivated by Mohar’s conjecture, Schaefer and Stefankovic provided an algorithm that computes a cross-cap drawing with a minimal number of cross-caps for a graph G such that each edge of the graph enters each cross-cap at most twice. In this talk, I will first outline a connection between cross-cap drawings and an algorithm coming from computational biology to compute the signed reversal distance between two permutations. This connection will then be leveraged to answer two computational problems on graphs embedded on surfaces. First, I show how to compute a “short” canonical decomposition for a non-orientable surface with a graph embedded on it. Such canonical decompositions were known for orientable surfaces, but the techniques used to compute them do not generalize to non-orientable surfaces due to their more complex nature. Second, I explain how to build a counter example to a stronger version of Mohar’s conjecture that is stated for pseudo-triangulations. This is joint work with Alfredo Hubard and Arnaud de Mesmay.

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ZOOM ID: 997 8258 4700(pw: 1234)

The qualitative theory of dynamical systems mainly provides a mathematical framework for analyzing the long-time behavior of systems without necessarily finding solutions for the given ODEs. The theory of dynamical systems could be related to deep learning problems from various perspectives such as approximation, optimization, generalization, and explainability. In this talk, we first introduce the qualitative theory of dynamical systems. Then, we present numerical results as the application of the qualitative theory of dynamical systems to deep learning problems.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.

EO strata are subvarieties in the moduli space of g-dimensional abelian varieties in characterstic p which classify points with given isomorphism type of p-torson subgroups. We are interested in how automorphism groups of points vary in supersingular EO strata. We show that when g is even and p>3, there is an open dense of the maximal supersingular EO stratum in which every point has automorphism group {\pm 1}, and prove Oor's conjecture in this case. This is joint work in progress with Valentijn Karemaker.

This lecture explores the topics and areas that have guided my research in computational mathematics and machine learning in recent years. Numerical methods in computational science are essential for comprehending real-world phenomena, and deep neural networks have achieved state-of-the-art results in a range of fields. The rapid expansion and outstanding success of deep learning and scientific computing have led to their applications across multiple disciplines, ranging from fluid dynamics to material sciences. In this lecture, I will focus on bridging machine learning with applied mathematics, specifically discussing topics such as scientific machine learning, numerical PDEs, and mathematical approaches of machine learning, including generative models and adversarial examples.

The Tomas-Stein inequality is a fundamental inequality in Fourier Analysis. It measures the L^4 norm of the Fourier transform of the sphere surface measure in terms of the L^2 norm. It is possible because the sphere has a positive Gaussian curvature. In this talk we will present what is an extremizer problem to this inequality and what is the progress of this problem.

We introduce a general equivalence problems for geometric structures arising from minimal rational curves on uniruled complex projective manifolds. To study these problems, we need approaches fusing differential geometry and algebraic geometry. Among such geometric structures, those associated to homogeneous manifolds are particularly accessible to differential-geometric methods of Cartan geometry. But even in these cases, only a few cases have been worked out so far. We review some recent developments.

In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture for digraphs. If G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most ⌈n/r⌉. In this talk, we prove that Aharoni’s conjecture holds up to an additive constant. Specifically, we show that for each fixed r, there exists a constant c such that if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most n/r+c. This is joint work with Patrick Hompe.

Deep learning techniques are increasingly applied to scientific problems, where the precision of networks is crucial. Despite being deemed as universal function approximators, neural networks, in practice, struggle to reduce the prediction errors below O(10−5) even with large network size and extended training iterations. To address this issue, we developed the multi-stage neural networks that divides the training process into different stages, with each stage using a new network that is optimized to fit the residue from the previous stage. Across successive stages, the residue magnitudes decreases substantially and follows an inverse power-law relationship with the residue frequencies. The multi-stage neural networks effectively mitigate the spectral biases associated with regular neural networks, enabling them to capture the high frequency feature of target functions. We demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision O(10−16) of double-floating point within a finite number of iterations. Such levels of accuracy are rarely attainable using single neural networks alone.

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ZOOM ID: 997 8258 4700(pw: 1234)

Link prediction (LP), inferring the connectivity between nodes, is a significant research area in graph data, where a link represents essential information on relationships between nodes. Although graph neural network (GNN)-based models have achieved high performance in LP, understanding why they perform well is challenging because most comprise complex neural networks. We employ persistent homology (PH), a topological data analysis method that helps analyze the topological information of graphs, to explain the reasons for the high performance. We propose a novel method that employs PH for LP (PHLP) focusing on how the presence or absence of target links influences the overall topology. The PHLP utilizes the angle hop subgraph and new node labeling called degree double radius node labeling (Degree DRNL), distinguishing the information of graphs better than DRNL. Using only a classifier, PHLP performs similarly to state-of-the-art (SOTA) models on most benchmark datasets. Incorporating the outputs calculated using PHLP into the existing GNN-based SOTA models improves performance across all benchmark datasets. To the best of our knowledge, PHLP is the first method of applying PH to LP without GNNs. The proposed approach, employing PH while not relying on neural networks, enables the identification of crucial factors for improving performance. https://arxiv.org/abs/2404.15225

This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.

최신 논문 리뷰: Rapid Convergence of Unadjusted Langevin Algorithm (Vempala et al) and Score-Based Generative Models(Song et al)

I tell a personal story of how a mathematician working in complex algebraic geometry had come to discover the relevance of Cartan geometry, a subject in differential geometry, in an old problem in algebraic geometry, the problem of deformations of Grassmannians as projective manifolds, which originated from the work of Kodaira and Spencer. In my joint work with Ngaiming Mok, we used the theory of minimal rational curves to study such deformations and it reduced the question to a problem in Cartan geometry.

This talk aims to consider the attainability of the Hardy-type inequality in the bounded smooth domain with average-zero type constraint. Since the criteria of the attainability depends to the concentration-compactness type arguments, we will briefly introduce the results for some classical Hardy-type inequalities and the concentration-compactness arguments. Subsequently, we propose new function spaces that well define the new inequalities. Finally, we will discuss the attainability of the optimal constant of the inequality in the general smooth domain.

In this talk, we will introduce support properties of solutions to nonlinear stochastic reaction-diffusion equations driven by random noise ˙W : ∂tu = aijuxixj + biuxi + cu + ξσ(u) ˙W , (ω, t, x) ∈ Ω × R+ × Rd; u(0, ·) = u0, where aij , bi, c and ξ are bounded and random coefficients. The noise ˙W is spacetime white noise or spatially homogeneous colored noise satisfying reinforced Dalang’s condition. We present examples of conditions on σ(u) that guarantee the compact support property of the solution. In addition, we suggest potential generalization of these conditions. This is joint work with Kunwoo Kim and Jaeyun Yi.

A family $\mathcal F$ of (di)graphs is said to have the half- or quarter-integral Erdős-Pósa property if, for any integer $k$ and any (di)graph $G$, there either exist $k$ copies of graphs in $\mathcal F$ within $G$ such that any vertex of $G$ is contained in at most 2, respectively at most 4, of these copies, or there exists a vertex set $A$ of size at most $f(k)$ such that $G - A$ contains no copies of graphs in $\mathcal F$. Very recently we showed that even dicycles have the quarter-integral Erdős-Pósa property [STOC'24] via the proof of a structure theorem for digraphs without large packings of even dicycles. In this talk we discuss our current effort to improve this approach towards the half-integral Erdős-Pósa property, which would be best possible, as even dicycles do not have the integral Erdős-Pósa property. Complementing the talk given by Sebastian Wiederrecht in this seminar regarding our initial result, we also shine a light on some of the particulars of the embedding we use in lieu of flatness and how this helps us to move even dicycles through the digraph. In the process of this, we highlight the parts of the proof that initially caused the result to be quarter-integral. (This is joint work with Ken-ichi Kawarabayashi, Stephan Kreutzer, and Sebastian Wiederrecht.)

In nonstationary bandit learning problems, the decision-maker must continually gather information and adapt their action selection as the latent state of the environment evolves. In each time period, some latent optimal action maximizes expected reward under the environment state. We view the optimal action sequence as a stochastic process, and take an information-theoretic approach to analyze attainable performance. We bound per-period regret in terms of the entropy rate of the optimal action process. The bound applies to a wide array of problems studied in the literature and reflects the problem’s information structure through its information-ratio.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.

Hamiltonian time-series data are observations derived from a Hamiltonian dynamical system. Our goal is to analyze the time-series data using the topological information of Hamiltonian dynamical systems. Exact Multi-parameter Persistent Homology is one aspect of this analysis, in this case, the Hamiltonian system is composed of uncoupled one-dimensional harmonic oscillators. This is a very simple model. However, we can induce the exact persistence barcode formula from it. From this formula, we can obtain a calculable and interpretable analysis. Filtration is necessary to extract the topological information of data and to define persistent homology. However, in many cases, we use static filtrations, such as the Vietoris-Rips filtration. My ongoing research is on topological optimization, which involves finding a filtration in Exact Multi-parameter Persistent Homology that minimizes the cross-entropy loss function for the classification of time-series data.

"An improved rhythmicity analysis method using Gaussian Processes detects cell-density dependent circadian oscillations in stem cells", ArXiv. (2023) will be discussed in this Journal Club. Detecting oscillations in time series remains a challenging problem even after decades of research. In chronobiology, rhythms in time series (for instance gene expression, eclosion, egg-laying and feeding) datasets tend to be low amplitude, display large variations amongst replicates, and often exhibit varying peak-to-peak distances (non-stationarity). Most currently available rhythm detection methods are not specifically designed to handle such datasets. Here we introduce a new method, ODeGP (Oscillation Detection using Gaussian Processes), which combines Gaussian Process (GP) regression with Bayesian inference to provide a flexible approach to the problem. Besides naturally incorporating measurement errors and non-uniformly sampled data, ODeGP uses a recently developed kernel to improve detection of non-stationary waveforms. An additional advantage is that by using Bayes factors instead of p-values, ODeGP models both the null (non-rhythmic) and the alternative (rhythmic) hypotheses. Using a variety of synthetic datasets we first demonstrate that ODeGP almost always outperforms eight commonly used methods in detecting stationary as well as non-stationary oscillations. Next, on analyzing existing qPCR datasets that exhibit low amplitude and noisy oscillations, we demonstrate that our method is more sensitive compared to the existing methods at detecting weak oscillations. Finally, we generate new qPCR time-series datasets on pluripotent mouse embryonic stem cells, which are expected to exhibit no oscillations of the core circadian clock genes. Surprisingly, we discover using ODeGP that increasing cell density can result in the rapid generation of oscillations in the Bmal1 gene, thus highlighting our method’s ability to discover unexpected patterns. In its current implementation, ODeGP (available as an R package) is meant only for analyzing single or a few time-trajectories, not genome-wide datasets. If you want to participate in the seminar, you need to enter IBS builiding (https://www.ibs.re.kr/bimag/visiting/). Please contact if you first come IBS to get permission to enter IBS building.

TBA

In recent years, ``stealthy'' particle systems have gained considerable attention in condensed matter physics. These are particle systems for which the diffraction spectrum or structure function (i.e. the Fourier transform of the truncated pair correlation function) vanishes in a neighbourhood of the origin in the wave space. These systems are believed to exhibit the phenomenon of ``cloaking'', i.e. being invisible to probes of certain frequencies. They also exhibit the phenomenon of hyperuniformity, namely suppressed fluctuations of particle counts, a property that has been shown to arise in a wide array of settings in chemistry, physics and biology. We will demonstrate that stealthy particle systems (and their natural extensions to stealthy stochastic processes) exhibit a highly rigid structure; in particular, their entropy per unit volume is degenerate, and any spatial void in such a system cannot exceed a certain size. Time permitting, we will also discuss the intriguing correlation geometry of such systems and its interplay with the analytical properties of their diffraction spectrum. Based on joint works with Joel Lebowitz and Kartick Adhikari.

We say that two functors Λ and Γ between thin categories of relational structures are adjoint if for all structures A and B, we have that Λ(A) maps homomorphically to B if and only if A maps homomorphically to Γ(B). If this is the case Λ is called the left adjoint to Γ and Γ the right adjoint to Λ. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We shall present several recent advances in this direction including a new approach based on the notion of Datalog Program borrowed from logic.

In this presentation, we discuss comprehensive frequency domain methods for estimating and inferring the second-order structure of spatial point processes. The main element here is on utilizing the discrete Fourier transform (DFT) of the point pattern and its tapered counterpart. Under second-order stationarity, we show that both the DFTs and the tapered DFTs are asymptotically jointly independent Gaussian even when the DFTs share the same limiting frequencies. Based on these results, we establish an α-mixing central limit theorem for a statistic formulated as a quadratic form of the tapered DFT. As applications, we derive the asymptotic distribution of the kernel spectral density estimator and establish a frequency domain inferential method for parametric stationary point processes. For the latter, the resulting model parameter estimator is computationally tractable and yields meaningful interpretations even in the case of model misspecification. We investigate the finite sample performance of our estimator through simulations, considering scenarios of both correctly specified and misspecified models. Joint work with Yongtao Guan @CUHK-Shenzhen.

The r-th cactus variety of a subvariety X in a projective space generalises secant variety of X and it is defined using linear spans of finite schemes of degree r. It's original purpose was to study the vanishing sets of catalecticant minors. We propose adding a scheme structure to the cactus variety and we define it via relative linear spans of families of finite schemes over a potentially non-reduced base. In this way we are able to study the vanishing scheme of the catalecticant minors. For X which is a sufficiently large Veronese reembedding of projective variety, we show that r-th cactus scheme and the zero scheme of appropriate catalecticant minors agree on an open and dense subset which is the complement of the (r-1)-st cactus variety/scheme. As an application, we can describe the singular locus of (in particular) secant varieties to high degree Veronese varieties. Based on a joint work with Hanieh Keneshlou.

"Phenotypic switching in gene regulatory networks", PNAS. (2014) will be discussed in this Journal Club. Noise in gene expression can lead to reversible phenotypic switching. Several experimental studies have shown that the abundance distributions of proteins in a population of isogenic cells may display multiple distinct maxima. Each of these maxima may be associated with a subpopulation of a particular phenotype, the quantification of which is important for understanding cellular decision-making. Here, we devise a methodology which allows us to quantify multimodal gene expression distributions and single-cell power spectra in gene regulatory networks. Extending the commonly used linear noise approximation, we rigorously show that, in the limit of slow promoter dynamics, these distributions can be systematically approximated as a mixture of Gaussian components in a wide class of networks. The resulting closed-form approximation provides a practical tool for studying complex nonlinear gene regulatory networks that have thus far been amenable only to stochastic simulation. We demonstrate the applicability of our approach in a number of genetic networks, uncovering previously unidentified dynamical characteristics associated with phenotypic switching. Specifically, we elucidate how the interplay of transcriptional and translational regulation can be exploited to control the multimodality of gene expression distributions in two-promoter networks. We demonstrate how phenotypic switching leads to birhythmical expression in a genetic oscillator, and to hysteresis in phenotypic induction, thus highlighting the ability of regulatory networks to retain memory. If you want to participate in the seminar, you need to enter IBS builiding (https://www.ibs.re.kr/bimag/visiting/). Please contact if you first come IBS to get permission to enter IBS building.

Delta-matroids are a generalization of matroids with connections to many parts of graph theory and combinatorics (such as matching theory and the structure of topological graph embeddings). Formally, a delta-matroid is a pair $D=(V,\mathcal F)$ where $\mathcal F$ is a collection of subsets of V known as "feasible sets." (They can be thought of as generalizing the set of bases of a matroid, while relaxing the condition that all bases must have the same cardinality.) Like with matroids, an important class of delta-matroids are linear delta-matroids, where the feasible sets are represented via a skew-symmetric matrix. Prominent examples of linear delta-matroids include linear matroids and matching delta-matroids (where the latter are represented via the famous Tutte matrix). However, the study of algorithms over delta-matroids seems to have been much less developed than over matroids. In this talk, we review recent results on representations of and algorithms over linear delta-matroids. We first focus on classical polynomial-time aspects. We present a new (equivalent) representation of linear delta-matroids that is more suitable for algorithmic purposes, and we show that so-called delta-sums and unions of linear delta-matroids are linear. As a result, we get faster (randomized) algorithms for Linear Delta-matroid Parity and Linear Delta-matroid Intersection, improving results from Geelen et al. (2004). We then move on to parameterized complexity aspects of linear delta-matroids. We find that many results regarding linear matroids which have had applications in FPT algorithms and kernelization directly generalize to linear delta-matroids of bounded rank. On the other hand, unlike with matroids, there is a significant difference between the "rank" and "cardinality" parameters - the structure of bounded-cardinality feasible sets in a delta-matroid of unbounded rank is significantly harder to deal with than feasible sets in a bounded-rank delta-matroid.

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ZOOM ID: 997 8258 4700(pw: 1234)

In the past decade, machine learning methods (MLMs) for solving partial differential equations (PDEs) have gained significant attention as a novel numerical approach. Indeed, a tremendous number of research projects have surged that apply MLMs to various applications, ranging from geophysics to biophysics. This surge in interest stems from the ability of MLMs to rapidly predict solutions for complex physical systems, even those involving multi-physics phenomena, uncertainty, and real-world data assimilation. This trend has led many to hopeful thinking MLMs as a potential game-changer in PDE solving. However, despite the hopeful thinking on MLMs, there are still significant challenges to overcome. These include limits compared to conventional numerical approaches, a lack of thorough analytical understanding of its accuracy, and the potentially long training times involved. In this talk, I will first assess the current state of MLMs for solving PDEs. Following this, we will explore what roles MLMs should play to become a conventional numerical scheme.

This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.

I will discuss some recent progress on the freeness problem for groups of 2x2 rational matrices generated by two parabolic matrices. In particular, I will discuss recent progress on determining the structural properties of such groups (beyond freeness) and when they have finite index in the finitely presented group SL(2,Z[1/m]), for appropriately chosen m.

Atoms and molecules aim to minimize surface energy, while crystals exhibit directional preferences. Starting from the classical isoperimetric problem, we investigate the evolution of volume-preserving crystalline mean curvature flow. Defining a notion of viscosity solutions, we demonstrate the preservation of geometric properties associated with the Wulff shape. We establish global-in-time existence and regularity for a class of initial data. Furthermore, we discuss recent findings on the long-time behavior of the flow towards the critical point of the anisotropic perimeter functional in a planar setting.

The positive discrepancy of a graph $G$ of edge density $p$ is defined as the maximum of $e(U) - p|U|(|U|-1)/2$, where the maximum is taken over subsets of vertices in G. In 1993 Alon proved that if G is a $d$-regular graph on $n$ vertices and $d = O(n^{1/9})$, then the positive discrepancy of $G$ is at least $c d^{1/2}n$ for some constant $c$. We extend this result by proving lower bounds for the positive discrepancy with average degree d when $d < (1/2 - \epsilon)n$. We prove that the same lower bound remains true when $d < n^(2/3)$, while in the ranges $n^{2/3} < d < n^{4/5}$ and $n^{4/5} < d < (1/2 - \epsilon)n$ we prove that the positive discrepancy is at least $n^2/d$ and $d^{1/4}n/log(n)$ respectively. Our proofs are based on semidefinite programming and linear algebraic techniques. Our results are tight when $d < n^{3/4}$, thus demonstrating a change in the behaviour around $d = n^{2/3}$ when a random graph no longer minimises the positive discrepancy. As a by-product, we also present lower bounds for the second largest eigenvalue of a $d$-regular graph when $d < (1/2 - \epsilon)n$, thus extending the celebrated Alon-Boppana theorem. This is joint work with Benjamin Sudakov and István Tomon.

In this talk, we focus on the global existence of volume-preserving mean curvature flows. In the isotropic case, leveraging the gradient flow framework, we demonstrate the convergence of solutions to a ball for star-shaped initial data. On the other hand, for anisotropic and crystalline flows, we establish the global-in-time existence for a class of initial data with the reflection property, utilizing explicit discrete-in-time approximation methods.

Using the invariant splitting principle, we construct an infinite family of exotic pairs of contractible 4-manifolds which survive one stabilization. We argue that some of them are potential candidates for surviving two stabilizations.

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ZOOM ID: 997 8258 4700(pw: 1234)

The size and complexity of recent deep learning models continue to increase exponentially, causing a serious amount of hardware overheads for training those models. Contrary to inference-only hardware, neural network training is very sensitive to computation errors; hence, training processors must support high-precision computation to avoid a large performance drop, severely limiting their processing efficiency. This talk will introduce a comprehensive design approach to arrive at an optimal training processor design. More specifically, the talk will discuss how we should make important design decisions for training processors in more depth, including i) hardware-friendly training algorithms, ii) optimal data formats, and iii) processor architecture for high precision and utilization.

This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.

We begin the first talk by introducing the concept of an h-principle that is mostly accessible through the two important methods. One of the methods is the convex integration that was successfully used by Mueller and Sverak and has been applied to many important PDEs. The other is the so-called Baire category method that was mainly studied by Dacorogna and Marcellini. We compare these methods in applying to a toy example.

In the second talk of the series, we exhibit several examples of application of convex integration to important PDE problems. In particular, we shall sketch some ideas of proof such as in the p-Laplace equation and its parabolic analogue, Euler-Lagrange equation of a polyconvex energy, gradient flow of a polyconvex energy and polyconvex elastodynamics.

TBA

After a brief review of the history, some applications of these models will be reviewed. This will include descriptions of rogue waves, tsunami propagation, internal waves and blood flow. Some of the theory emanaging from these applications will then be sketched.

One of the classical and most fascinating problems at the intersection between combinatorics and number theory is the study of the parity of the partition function. Even though p(n) in widely believed to be equidistributed modulo 2, progress in the area has proven exceptionally hard. The best results available today, obtained incrementally over several decades by Serre, Soundarajan, Ono and many otehrs, do not even guarantee that, asymptotically, p(n) is odd for /sqrt{x} values of n/neq x, In this talk, we present a new, general conjectural framework that naturally places the parity of p(n) into the much broader, number-theoretic context of eta-eqotients. We discuss the history of this problem as well as recent progress on our "master conjecture," which includes novel results on multi-and regular partitions. We then show how seemingly unrelated classes of eta-equotients carry surprising (and surprisingly deep) connections modulo 2 to the partition function. One instance is the following striking result: If any t-multiparition function, with t/neq 0(mod 3), is odd with positive density, then so is p(n). (Note that proving either fact unconditionally seems entirely out of reach with current methods.) Throughout this talk, we will give a sense of the many interesting mathematical techniques that come into play in this area. They will include a variety of algebraic and combinatorial ideas, as well as tools from modular forms and number theory.

In this talk, we consider some polynomials which define Gaussian Graphical models in algebraic statistics. First, we briefly introduce background materials and some preliminary on this topic. Next, we regard a conjecture due to Sturmfels and Uhler concerning generation of the prime ideal of the variety associated to the Gaussian graphical model of any cycle graph and explain how to prove it. We also report a result on linear syzygies of any model coming from block graphs. The former work was done jointly with A. Conner and M. Michalek and the latter with J. Choe.

Bollobás proved that for every $k$ and $\ell$ such that $k\mathbb{Z}+\ell$ contains an even number, an $n$-vertex graph containing no cycle of length $\ell \bmod k$ can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs $\ell$ and $k$. We precisely determine the maximum number of edges in a graph containing no cycle of length $0 \bmod 4$. This is joint work with Ervin Győri, Binlong Li, Nika Salia, Kitti Varga and Manran Zhu.

We introduce bordered Floer theory and its involutive version, as well as their applications to knot complements. We will sketch the proof that invariant splittings of CFK and those of CFD correspond to each other under the Lipshitz-Ozsvath-Thurston correspondence, via invariant splitting principle, which is an ongoing work with Gary Guth.